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Theorem wpthswwlks2on 29981
Description: For two different vertices, a walk of length 2 between these vertices is a simple path of length 2 between these vertices in a simple graph. (Contributed by Alexander van der Vekens, 2-Mar-2018.) (Revised by AV, 13-May-2021.) (Revised by AV, 16-Mar-2022.)
Assertion
Ref Expression
wpthswwlks2on ((𝐺 ∈ USGraph ∧ 𝐴𝐵) → (𝐴(2 WSPathsNOn 𝐺)𝐵) = (𝐴(2 WWalksNOn 𝐺)𝐵))

Proof of Theorem wpthswwlks2on
Dummy variables 𝑓 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 wwlknon 29877 . . . . . . 7 (𝑤 ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ↔ (𝑤 ∈ (2 WWalksN 𝐺) ∧ (𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵))
21a1i 11 . . . . . 6 ((𝐺 ∈ USGraph ∧ 𝐴𝐵) → (𝑤 ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ↔ (𝑤 ∈ (2 WWalksN 𝐺) ∧ (𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵)))
32anbi1d 631 . . . . 5 ((𝐺 ∈ USGraph ∧ 𝐴𝐵) → ((𝑤 ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ∧ ∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤) ↔ ((𝑤 ∈ (2 WWalksN 𝐺) ∧ (𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵) ∧ ∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤)))
4 3anass 1095 . . . . . . 7 ((𝑤 ∈ (2 WWalksN 𝐺) ∧ (𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵) ↔ (𝑤 ∈ (2 WWalksN 𝐺) ∧ ((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵)))
54anbi1i 624 . . . . . 6 (((𝑤 ∈ (2 WWalksN 𝐺) ∧ (𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵) ∧ ∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤) ↔ ((𝑤 ∈ (2 WWalksN 𝐺) ∧ ((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵)) ∧ ∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤))
6 anass 468 . . . . . 6 (((𝑤 ∈ (2 WWalksN 𝐺) ∧ ((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵)) ∧ ∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤) ↔ (𝑤 ∈ (2 WWalksN 𝐺) ∧ (((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵) ∧ ∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤)))
75, 6bitri 275 . . . . 5 (((𝑤 ∈ (2 WWalksN 𝐺) ∧ (𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵) ∧ ∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤) ↔ (𝑤 ∈ (2 WWalksN 𝐺) ∧ (((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵) ∧ ∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤)))
83, 7bitrdi 287 . . . 4 ((𝐺 ∈ USGraph ∧ 𝐴𝐵) → ((𝑤 ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ∧ ∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤) ↔ (𝑤 ∈ (2 WWalksN 𝐺) ∧ (((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵) ∧ ∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤))))
98rabbidva2 3438 . . 3 ((𝐺 ∈ USGraph ∧ 𝐴𝐵) → {𝑤 ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ∣ ∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤} = {𝑤 ∈ (2 WWalksN 𝐺) ∣ (((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵) ∧ ∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤)})
10 usgrupgr 29202 . . . . . . . . . . 11 (𝐺 ∈ USGraph → 𝐺 ∈ UPGraph)
11 wlklnwwlknupgr 29906 . . . . . . . . . . 11 (𝐺 ∈ UPGraph → (∃𝑓(𝑓(Walks‘𝐺)𝑤 ∧ (♯‘𝑓) = 2) ↔ 𝑤 ∈ (2 WWalksN 𝐺)))
1210, 11syl 17 . . . . . . . . . 10 (𝐺 ∈ USGraph → (∃𝑓(𝑓(Walks‘𝐺)𝑤 ∧ (♯‘𝑓) = 2) ↔ 𝑤 ∈ (2 WWalksN 𝐺)))
1312bicomd 223 . . . . . . . . 9 (𝐺 ∈ USGraph → (𝑤 ∈ (2 WWalksN 𝐺) ↔ ∃𝑓(𝑓(Walks‘𝐺)𝑤 ∧ (♯‘𝑓) = 2)))
1413adantr 480 . . . . . . . 8 ((𝐺 ∈ USGraph ∧ 𝐴𝐵) → (𝑤 ∈ (2 WWalksN 𝐺) ↔ ∃𝑓(𝑓(Walks‘𝐺)𝑤 ∧ (♯‘𝑓) = 2)))
15 simprl 771 . . . . . . . . . . . . . 14 ((((𝐺 ∈ USGraph ∧ 𝐴𝐵) ∧ ((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵)) ∧ (𝑓(Walks‘𝐺)𝑤 ∧ (♯‘𝑓) = 2)) → 𝑓(Walks‘𝐺)𝑤)
16 simprl 771 . . . . . . . . . . . . . . 15 (((𝐺 ∈ USGraph ∧ 𝐴𝐵) ∧ ((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵)) → (𝑤‘0) = 𝐴)
1716adantr 480 . . . . . . . . . . . . . 14 ((((𝐺 ∈ USGraph ∧ 𝐴𝐵) ∧ ((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵)) ∧ (𝑓(Walks‘𝐺)𝑤 ∧ (♯‘𝑓) = 2)) → (𝑤‘0) = 𝐴)
18 fveq2 6906 . . . . . . . . . . . . . . . 16 ((♯‘𝑓) = 2 → (𝑤‘(♯‘𝑓)) = (𝑤‘2))
1918ad2antll 729 . . . . . . . . . . . . . . 15 ((((𝐺 ∈ USGraph ∧ 𝐴𝐵) ∧ ((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵)) ∧ (𝑓(Walks‘𝐺)𝑤 ∧ (♯‘𝑓) = 2)) → (𝑤‘(♯‘𝑓)) = (𝑤‘2))
20 simprr 773 . . . . . . . . . . . . . . . 16 (((𝐺 ∈ USGraph ∧ 𝐴𝐵) ∧ ((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵)) → (𝑤‘2) = 𝐵)
2120adantr 480 . . . . . . . . . . . . . . 15 ((((𝐺 ∈ USGraph ∧ 𝐴𝐵) ∧ ((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵)) ∧ (𝑓(Walks‘𝐺)𝑤 ∧ (♯‘𝑓) = 2)) → (𝑤‘2) = 𝐵)
2219, 21eqtrd 2777 . . . . . . . . . . . . . 14 ((((𝐺 ∈ USGraph ∧ 𝐴𝐵) ∧ ((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵)) ∧ (𝑓(Walks‘𝐺)𝑤 ∧ (♯‘𝑓) = 2)) → (𝑤‘(♯‘𝑓)) = 𝐵)
23 eqid 2737 . . . . . . . . . . . . . . . . . . . . . 22 (Vtx‘𝐺) = (Vtx‘𝐺)
2423wlkp 29634 . . . . . . . . . . . . . . . . . . . . 21 (𝑓(Walks‘𝐺)𝑤𝑤:(0...(♯‘𝑓))⟶(Vtx‘𝐺))
25 oveq2 7439 . . . . . . . . . . . . . . . . . . . . . 22 ((♯‘𝑓) = 2 → (0...(♯‘𝑓)) = (0...2))
2625feq2d 6722 . . . . . . . . . . . . . . . . . . . . 21 ((♯‘𝑓) = 2 → (𝑤:(0...(♯‘𝑓))⟶(Vtx‘𝐺) ↔ 𝑤:(0...2)⟶(Vtx‘𝐺)))
2724, 26syl5ibcom 245 . . . . . . . . . . . . . . . . . . . 20 (𝑓(Walks‘𝐺)𝑤 → ((♯‘𝑓) = 2 → 𝑤:(0...2)⟶(Vtx‘𝐺)))
2827imp 406 . . . . . . . . . . . . . . . . . . 19 ((𝑓(Walks‘𝐺)𝑤 ∧ (♯‘𝑓) = 2) → 𝑤:(0...2)⟶(Vtx‘𝐺))
29 id 22 . . . . . . . . . . . . . . . . . . . . 21 (𝑤:(0...2)⟶(Vtx‘𝐺) → 𝑤:(0...2)⟶(Vtx‘𝐺))
30 2nn0 12543 . . . . . . . . . . . . . . . . . . . . . 22 2 ∈ ℕ0
31 0elfz 13664 . . . . . . . . . . . . . . . . . . . . . 22 (2 ∈ ℕ0 → 0 ∈ (0...2))
3230, 31mp1i 13 . . . . . . . . . . . . . . . . . . . . 21 (𝑤:(0...2)⟶(Vtx‘𝐺) → 0 ∈ (0...2))
3329, 32ffvelcdmd 7105 . . . . . . . . . . . . . . . . . . . 20 (𝑤:(0...2)⟶(Vtx‘𝐺) → (𝑤‘0) ∈ (Vtx‘𝐺))
34 nn0fz0 13665 . . . . . . . . . . . . . . . . . . . . . . 23 (2 ∈ ℕ0 ↔ 2 ∈ (0...2))
3530, 34mpbi 230 . . . . . . . . . . . . . . . . . . . . . 22 2 ∈ (0...2)
3635a1i 11 . . . . . . . . . . . . . . . . . . . . 21 (𝑤:(0...2)⟶(Vtx‘𝐺) → 2 ∈ (0...2))
3729, 36ffvelcdmd 7105 . . . . . . . . . . . . . . . . . . . 20 (𝑤:(0...2)⟶(Vtx‘𝐺) → (𝑤‘2) ∈ (Vtx‘𝐺))
3833, 37jca 511 . . . . . . . . . . . . . . . . . . 19 (𝑤:(0...2)⟶(Vtx‘𝐺) → ((𝑤‘0) ∈ (Vtx‘𝐺) ∧ (𝑤‘2) ∈ (Vtx‘𝐺)))
3928, 38syl 17 . . . . . . . . . . . . . . . . . 18 ((𝑓(Walks‘𝐺)𝑤 ∧ (♯‘𝑓) = 2) → ((𝑤‘0) ∈ (Vtx‘𝐺) ∧ (𝑤‘2) ∈ (Vtx‘𝐺)))
40 eleq1 2829 . . . . . . . . . . . . . . . . . . 19 ((𝑤‘0) = 𝐴 → ((𝑤‘0) ∈ (Vtx‘𝐺) ↔ 𝐴 ∈ (Vtx‘𝐺)))
41 eleq1 2829 . . . . . . . . . . . . . . . . . . 19 ((𝑤‘2) = 𝐵 → ((𝑤‘2) ∈ (Vtx‘𝐺) ↔ 𝐵 ∈ (Vtx‘𝐺)))
4240, 41bi2anan9 638 . . . . . . . . . . . . . . . . . 18 (((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵) → (((𝑤‘0) ∈ (Vtx‘𝐺) ∧ (𝑤‘2) ∈ (Vtx‘𝐺)) ↔ (𝐴 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺))))
4339, 42imbitrid 244 . . . . . . . . . . . . . . . . 17 (((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵) → ((𝑓(Walks‘𝐺)𝑤 ∧ (♯‘𝑓) = 2) → (𝐴 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺))))
4443adantl 481 . . . . . . . . . . . . . . . 16 (((𝐺 ∈ USGraph ∧ 𝐴𝐵) ∧ ((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵)) → ((𝑓(Walks‘𝐺)𝑤 ∧ (♯‘𝑓) = 2) → (𝐴 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺))))
4544imp 406 . . . . . . . . . . . . . . 15 ((((𝐺 ∈ USGraph ∧ 𝐴𝐵) ∧ ((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵)) ∧ (𝑓(Walks‘𝐺)𝑤 ∧ (♯‘𝑓) = 2)) → (𝐴 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺)))
46 vex 3484 . . . . . . . . . . . . . . . 16 𝑓 ∈ V
47 vex 3484 . . . . . . . . . . . . . . . 16 𝑤 ∈ V
4846, 47pm3.2i 470 . . . . . . . . . . . . . . 15 (𝑓 ∈ V ∧ 𝑤 ∈ V)
4923iswlkon 29675 . . . . . . . . . . . . . . 15 (((𝐴 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺)) ∧ (𝑓 ∈ V ∧ 𝑤 ∈ V)) → (𝑓(𝐴(WalksOn‘𝐺)𝐵)𝑤 ↔ (𝑓(Walks‘𝐺)𝑤 ∧ (𝑤‘0) = 𝐴 ∧ (𝑤‘(♯‘𝑓)) = 𝐵)))
5045, 48, 49sylancl 586 . . . . . . . . . . . . . 14 ((((𝐺 ∈ USGraph ∧ 𝐴𝐵) ∧ ((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵)) ∧ (𝑓(Walks‘𝐺)𝑤 ∧ (♯‘𝑓) = 2)) → (𝑓(𝐴(WalksOn‘𝐺)𝐵)𝑤 ↔ (𝑓(Walks‘𝐺)𝑤 ∧ (𝑤‘0) = 𝐴 ∧ (𝑤‘(♯‘𝑓)) = 𝐵)))
5115, 17, 22, 50mpbir3and 1343 . . . . . . . . . . . . 13 ((((𝐺 ∈ USGraph ∧ 𝐴𝐵) ∧ ((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵)) ∧ (𝑓(Walks‘𝐺)𝑤 ∧ (♯‘𝑓) = 2)) → 𝑓(𝐴(WalksOn‘𝐺)𝐵)𝑤)
52 simplll 775 . . . . . . . . . . . . . 14 ((((𝐺 ∈ USGraph ∧ 𝐴𝐵) ∧ ((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵)) ∧ (𝑓(Walks‘𝐺)𝑤 ∧ (♯‘𝑓) = 2)) → 𝐺 ∈ USGraph)
53 simprr 773 . . . . . . . . . . . . . 14 ((((𝐺 ∈ USGraph ∧ 𝐴𝐵) ∧ ((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵)) ∧ (𝑓(Walks‘𝐺)𝑤 ∧ (♯‘𝑓) = 2)) → (♯‘𝑓) = 2)
54 simpllr 776 . . . . . . . . . . . . . 14 ((((𝐺 ∈ USGraph ∧ 𝐴𝐵) ∧ ((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵)) ∧ (𝑓(Walks‘𝐺)𝑤 ∧ (♯‘𝑓) = 2)) → 𝐴𝐵)
55 usgr2wlkspth 29779 . . . . . . . . . . . . . 14 ((𝐺 ∈ USGraph ∧ (♯‘𝑓) = 2 ∧ 𝐴𝐵) → (𝑓(𝐴(WalksOn‘𝐺)𝐵)𝑤𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤))
5652, 53, 54, 55syl3anc 1373 . . . . . . . . . . . . 13 ((((𝐺 ∈ USGraph ∧ 𝐴𝐵) ∧ ((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵)) ∧ (𝑓(Walks‘𝐺)𝑤 ∧ (♯‘𝑓) = 2)) → (𝑓(𝐴(WalksOn‘𝐺)𝐵)𝑤𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤))
5751, 56mpbid 232 . . . . . . . . . . . 12 ((((𝐺 ∈ USGraph ∧ 𝐴𝐵) ∧ ((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵)) ∧ (𝑓(Walks‘𝐺)𝑤 ∧ (♯‘𝑓) = 2)) → 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤)
5857ex 412 . . . . . . . . . . 11 (((𝐺 ∈ USGraph ∧ 𝐴𝐵) ∧ ((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵)) → ((𝑓(Walks‘𝐺)𝑤 ∧ (♯‘𝑓) = 2) → 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤))
5958eximdv 1917 . . . . . . . . . 10 (((𝐺 ∈ USGraph ∧ 𝐴𝐵) ∧ ((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵)) → (∃𝑓(𝑓(Walks‘𝐺)𝑤 ∧ (♯‘𝑓) = 2) → ∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤))
6059ex 412 . . . . . . . . 9 ((𝐺 ∈ USGraph ∧ 𝐴𝐵) → (((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵) → (∃𝑓(𝑓(Walks‘𝐺)𝑤 ∧ (♯‘𝑓) = 2) → ∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤)))
6160com23 86 . . . . . . . 8 ((𝐺 ∈ USGraph ∧ 𝐴𝐵) → (∃𝑓(𝑓(Walks‘𝐺)𝑤 ∧ (♯‘𝑓) = 2) → (((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵) → ∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤)))
6214, 61sylbid 240 . . . . . . 7 ((𝐺 ∈ USGraph ∧ 𝐴𝐵) → (𝑤 ∈ (2 WWalksN 𝐺) → (((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵) → ∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤)))
6362imp 406 . . . . . 6 (((𝐺 ∈ USGraph ∧ 𝐴𝐵) ∧ 𝑤 ∈ (2 WWalksN 𝐺)) → (((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵) → ∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤))
6463pm4.71d 561 . . . . 5 (((𝐺 ∈ USGraph ∧ 𝐴𝐵) ∧ 𝑤 ∈ (2 WWalksN 𝐺)) → (((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵) ↔ (((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵) ∧ ∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤)))
6564bicomd 223 . . . 4 (((𝐺 ∈ USGraph ∧ 𝐴𝐵) ∧ 𝑤 ∈ (2 WWalksN 𝐺)) → ((((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵) ∧ ∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤) ↔ ((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵)))
6665rabbidva 3443 . . 3 ((𝐺 ∈ USGraph ∧ 𝐴𝐵) → {𝑤 ∈ (2 WWalksN 𝐺) ∣ (((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵) ∧ ∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤)} = {𝑤 ∈ (2 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵)})
679, 66eqtrd 2777 . 2 ((𝐺 ∈ USGraph ∧ 𝐴𝐵) → {𝑤 ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ∣ ∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤} = {𝑤 ∈ (2 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵)})
6823iswspthsnon 29876 . 2 (𝐴(2 WSPathsNOn 𝐺)𝐵) = {𝑤 ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ∣ ∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤}
6923iswwlksnon 29873 . 2 (𝐴(2 WWalksNOn 𝐺)𝐵) = {𝑤 ∈ (2 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵)}
7067, 68, 693eqtr4g 2802 1 ((𝐺 ∈ USGraph ∧ 𝐴𝐵) → (𝐴(2 WSPathsNOn 𝐺)𝐵) = (𝐴(2 WWalksNOn 𝐺)𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1087   = wceq 1540  wex 1779  wcel 2108  wne 2940  {crab 3436  Vcvv 3480   class class class wbr 5143  wf 6557  cfv 6561  (class class class)co 7431  0cc0 11155  2c2 12321  0cn0 12526  ...cfz 13547  chash 14369  Vtxcvtx 29013  UPGraphcupgr 29097  USGraphcusgr 29166  Walkscwlks 29614  WalksOncwlkson 29615  SPathsOncspthson 29733   WWalksN cwwlksn 29846   WWalksNOn cwwlksnon 29847   WSPathsNOn cwwspthsnon 29849
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-ac2 10503  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-ifp 1064  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-tp 4631  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-2o 8507  df-oadd 8510  df-er 8745  df-map 8868  df-pm 8869  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-dju 9941  df-card 9979  df-ac 10156  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-2 12329  df-3 12330  df-n0 12527  df-xnn0 12600  df-z 12614  df-uz 12879  df-fz 13548  df-fzo 13695  df-hash 14370  df-word 14553  df-concat 14609  df-s1 14634  df-s2 14887  df-s3 14888  df-edg 29065  df-uhgr 29075  df-upgr 29099  df-umgr 29100  df-uspgr 29167  df-usgr 29168  df-wlks 29617  df-wlkson 29618  df-trls 29710  df-trlson 29711  df-pths 29734  df-spths 29735  df-pthson 29736  df-spthson 29737  df-wwlks 29850  df-wwlksn 29851  df-wwlksnon 29852  df-wspthsnon 29854
This theorem is referenced by:  usgr2wspthons3  29984  frgr2wsp1  30349
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